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Metamath Proof Explorer

Theorem resmptd

Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypothesis resmptd.b ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion resmptd ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 resmptd.b ⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
2 resmpt ⊢ ( 𝐵 ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )
3 1 2 syl ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )